#7516114

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#45093	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2081 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14,19,24)(15,20,28,37)(16,21,17,22)(18,23,33,42)(25,34,30,39)(26,35,43,44)(27,36,32,41)(29,38,31,40);(1,2)(3,4)(6,9)(10,12)(13,15,19,28)(14,20,24,37)(16,25,17,30)(18,32,33,27)(21,34,22,39)(23,41,42,36)(26,29,43,31)(35,38,44,40);(2,4)(13,16,19,17)(14,21,24,22)(15,25,28,30)(18,29,33,31)(20,34,37,39)(23,38,42,40)(26,32,43,27)(35,41,44,36);(2,4)(5,7,8,11)(6,10,9,12)(13,17,19,16)(14,22,24,21)(15,26,28,43)(18,29,33,31)(20,35,37,44)(23,38,42,40)(25,32,30,27)(34,41,39,36);(6,9)(10,12)(16,17)(21,22)(25,30)(26,43)(29,31)(34,39)(35,44)(38,40)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,e^d=ei,h^d=hi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7423310);